Combination problem with repetition. GRE Anagram Grid questions. Why do we divide?

Question

If 7 people board an airport shuttle with only 3 available seats, how many different seating arrangement are possible? (Assume that 3 of the 7 will actually take the seats.) So I think the answer is 7! / 4! = 210 because the 7! is equal to the total number of combinations right? The 4! represents the combinations of seated people. So, the reason why I think the answer is 7! / 4! is because we need to uncount the different arrangements of the seated people because the seated people are all equivalent and indistinguishable; i.e., it doesn't matter how they are seated. But why aren't we subtracting? Why are we dividing?

Answer 1

Subtraction is done. The formula (look it up!) results in

7! / (7! - 3!)